Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)] \cdot [5(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)$ ) has angle $\frac{7}{4}\pi$ and radius $1$ The second number ( $5(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))$ ) has angle $\frac{1}{6}\pi$ and radius $5$ The radius of the result will be $1 \cdot 5$ , which is $5$ The angle of the result is $\frac{7}{4}\pi + \frac{1}{6}\pi = \frac{23}{12}\pi$ The radius of the result is $5$ and the angle of the result is $\frac{23}{12}\pi$.